In the International Grandmaster Championship, a total of 105 games were played where each team played exactly one game with all the other teams. How many teams were participating in the Championship?

15 teams

Step-by-step explanation:

Total number of games played in the tournament = 105

Number of participating teams = p

Each team plays against every other team , then the total number of games played by each team = (p - 1) , since a team can't play itself

To obtain total number of games played =

(Number of teams × number of games played) [p × (p-1)]

However, the teams played against each other only once :

Therefore, total number of games played:

[p × (p-1)] / 2

Since we've been given the total number of games played :

[p × (p-1)] / 2 = 105

p × (p-1) = 105 × 2

p × (p-1) = 210

p^2 - p = 210

p^2 - p - 210 = 0

p^2 - 15p + 14p - 210 = 0

p(p - 15) +14(p-15) = 0

(p - 15) = 0 or (p + 14) = 0

p = 15 or p = - 14

Number of teams can't be negative

Therefore p = 15

Number of teams = 15